Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Example 8.1.25
Use an iterated triple integral to obtain the volume of R, the region bounded above by the sphere x2+y2+z2=6, and below by the paraboloid z=x2+y2.
Solution
Mathematical Solution
Figure 8.1.25(a) shows the solid whose volume is obtained by iterating a triple integral in cylindrical coordinates in the order dz dr dθ.
∫02 π∫02∫r26− r2r dz dr dθ = 2 π 26−113
The sphere and paraboloid intersect in a circle of radius 2, lying in the plane z=2. This is determined by eliminating x2+y2 from the equations for the sphere and paraboloid, and solving the resulting equation z2+z=6 for z=2, in which case, z=x2+y2=r2=2.
Figure 8.1.25(a) Sphere and paraboloid
Maple Solution - Interactive
Table 8.1.25(a) provides a solution by a task template that integrates in cylindrical coordinates and draws the region of integration.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
Evaluate ∭RΨr,θ,z dv and Graph R
Volume Element dv
r dz dr dθ
r dz dθ dr
r dr dθ dz
r dr dz dθ
r dθ dr dz
r dθ dz dr
, where Ψ=
F=
G=
b=
f=
g=
a=
Table 8.1.24(a) Task template integrating in cylindrical coordinates
Since the iteration order can be taken as dz dr dθ, the task template in Table 8.1.25(a), using the MultiInt command from the Student MultivariateCalculus package, applies.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical
Iterated Triple Integral in Cylindrical Coordinates
Integrand:
1
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
r2
z2r,θ
6−r2
−r2+6
r1θ
0
r2θ
2
a
b
2 π
2π
Inert Integral: dz dr dθ
(Note automatic insertion of Jacobian.)
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=integral
∫02π∫02∫r2−r2+6rⅆzⅆrⅆθ
Value:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
−223π+432π
Stepwise Evaluation:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=steps
∫02π∫02∫r2−r2+6rⅆzⅆrⅆθ=∫02π∫02rzz=r2..−r2+6|rzz=r2..−r2+6ⅆrⅆθ=∫02π∫02r−r2+6−r2ⅆrⅆθ=∫02π−r44−−r2+6323r=0..2|−r44−−r2+6323r=0..2ⅆθ=∫02π−113+232ⅆθ=−113+232θθ=0..2π|−113+232θθ=0..2π
Table 8.1.24(b) Task template implementing the MultiInt command iterating in the order dz dr dθ
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Access the MultiInt command via the Context Panel
Type the integrand, 1.
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated Fill in the fields of the two dialogs shown below.
Context Panel: Evaluate Integral
1→MultiInt∫02π∫02∫r2−r2+6rⅆzⅆrⅆθ=42π3−223π
Table 8.1.25(c) provides a solution from first principles.
Calculus Palette: Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫02 π∫02∫r26− r2r ⅆz ⅆr ⅆθ = −223π+432π
Table 8.1.25(c) Integration via first principles
Maple Solution - Coded
Table 8.1.25(d) obtains a solution via the MultiInt command in the Student MultivariateCalculus package. See Table 8.1.25(b) for an implementation of the integration via a task template.
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
MultiInt1,z=r2..6−r2,r=0..2,θ=0..2 π,coordinates=cylindricalr,θ,z,output=integral
MultiInt1,z=r2..6−r2,r=0..2,θ=0..2 π,coordinates=cylindricalr,θ,z
423π−223π
Table 8.1.25(d) MultiInt command iterating in cylindrical coordinates in the order dz dr dθ
Table 8.1.25(e) implements the iterated integration via the top-level Int and int commands.
Intr,z=r2..6−r2,r=0..2,θ=0..2 π=intr,z=r2..6−r2,r=0..2,θ=0..2 π
∫02π∫02∫r2−r2+6rⅆzⅆrⅆθ=2−113+232π
Table 8.1.25(e) Top-level Int and int commands
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